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hw533 - PHYS 507 Homework V Assigned Friday Due Monday at...

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PHYS 507 Homework V Assigned: December 12, 2003, Friday. Due: December 22, 2003, Monday, at 5:00 pm. 1. You know that a classical magnetic dipole moment in a magnetic field ~ B has energy - · ~ B and it feels a torque × ~ B . In quantum mechanics we use the same Hamiltonian (energy) expression, but the expression for the torque is valid as well. Show that d dt D ~ S E t = h i t × ~ B for any uniform magnetic field ~ B where ~ S is the spin of any particle (not necessarily spin 1/2). [ You may assume = k ~ S for some constant k , although this is unnecessary.] 2. Let ~ S 1 and ~ S 2 denote the spins of two spin 1/2 particles and ~ S = ~ S 1 + ~ S 2 denote the total spin. We know that a rotation around x axis by an angle θ of the whole two-particle system is given by D ( θ, ˆ x ) = exp - i ¯ h θS x = exp - i ¯ h θS 1 x exp - i ¯ h θS 2 x = D 1 ( θ, ˆ x ) D 2 ( θ, ˆ x ) in other words we do rotate both particles independently and the result is the rotation of the whole.
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